Problem: Find the distance between the point ${(3, 5)}$ and the line $\enspace {y = -3x - 6}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Answer: First, find the equation of the perpendicular line that passes through ${(3, 5)}$ The slope of the blue line is ${-3}$ , and its negative reciprocal is ${\dfrac{1}{3}}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = \dfrac{1}{3}x + b}\thinspace$ We can plug our point, ${(3, 5)}$ , into this equation to solve for ${b}$ , the y-intercept. $5 = {\dfrac{1}{3}}(3) + {b}$ $5 = 1 + {b}$ $5 - 1 = {b} = 4$ The equation of the perpendicular line is $\enspace {y = \dfrac{1}{3}x + 4}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(-3, 3)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(3, 5)}$ and ${(-3, 3)}$ gives us: $\sqrt{( {3} - {-3} )^2 + ( {5} - {3} )^2}$ $= \sqrt{( 6 )^2 + ( 2 )^2} = \sqrt{40} = 2\sqrt{10}$ The distance between the point ${(3, 5)}$ and the line $\thinspace {y = -3x - 6}\enspace$ is $\thinspace2\sqrt{10}$.